Nonlinearity & Functional Analysis: Lectures on Nonlinear Problems in Mathematical Analysis

Nonlinearity and Functional Analysis: Lectures on Nonlinear Problems in Mathematical Analysis......Page 4
Copyright Page......Page 5
Contents......Page 8
Preface......Page 14
Notation and Terminology......Page 18
Suggestions for The Reader......Page 20
PART I: PRELIMINARIES......Page 22
1.1 How Nonlinear Problems Arise......Page 24
1.2 Typical Difficulties Encountered......Page 39
1.3 Facts from Functional Analysis......Page 46
1.4 Inequalities and Estimates......Page 60
1.5 Classical and Generalized Solutions of Differential Systems......Page 68
1.6 Mappings between Finite-Dimensional Spaces......Page 72
2.1 Elementary Calculus......Page 85
2.2 Specific Nonlinear Operators......Page 97
2.3 Analytic Operators......Page 105
2.4 Compact Operators......Page 109
2.5 Gradient Mappings......Page 114
2.6 Nonlinear Fredholm Operators......Page 120
2.7 Proper Mappings......Page 123
Notes......Page 128
PART II: LOCAL ANALYSIS......Page 130
3.1 Successive Approximations......Page 132
3.2 The Steepest Descent Method for Gradient Mappings......Page 148
3.3 Analytic Operators and the Majorant Method......Page 154
3.4 Generalized Inverse Function Theorems......Page 158
Notes......Page 166
4.1 Bifurcation Theory–A Constructive Approach......Page 170
4.2 Transcendental Methods in Bifurcation Theory......Page 184
4.3 Specific Bifurcation Phenomena......Page 194
4.4 Asymptotic Expansions and Singular Perturbations......Page 214
4.5 Some Singular Perturbation Problems of Classical Mathematical Physics......Page 225
Notes......Page 232
PART III: ANALYSIS IN THE LARGE......Page 236
5.1 Linearization......Page 238
5.2 Finite-Dimensional Approximations......Page 252
5.3 Homotopy, the Degree of Mappings, and Its Generalizations......Page 264
5.4 Homotopy and Mapping Properties of Nonlinear Operators......Page 287
5.5 Applications to Nonlinear Boundary Value Problems......Page 304
Notes......Page 317
6.1 Minimization Problems......Page 320
6.2 Specific Minimization Problems from Geometry and Physics......Page 334
6.3 lsoperimetric Problems......Page 345
6.4 Isoperimetric Problems in Geometry and Physics......Page 358
6.5 Critical Point Theory of Marston Morse in Hilbert Space......Page 374
6.6 The Critical Point Theory of Ljusternik and Schnirelmann......Page 387
6.7 Applications of the General Critical Point Theories......Page 396
Notes......Page 409
Appendix A. On Differentiable Manifolds......Page 412
Appendix B. On the Hodge-Kodaira Decomposition for Differential Forms......Page 417
References......Page 420
Index......Page 430
Pure and Applied Mathematics......Page 439